D.e inhomogneous non constant coeff

The question is find the solution of:

clearly the coefficients in what i would write down to be the auxilliary equation are not constants. Ive been told by my solution to use y= but i dont see how this gives(in the solutions)
giving m=4 or m=-2. and the corrosponding complementary function. Does anyone have any ideas or could show me how the method actually cancels out the non constant coefficeints? thanks. oh i think you are supposed to use liebnitz?? not sure

clearly the coefficients in what i would write down to be the auxilliary equation are not constants. Ive been told by my solution to use y= but i dont see how this gives(in the solutions)
giving m=4 or m=-2. and the corrosponding complementary function. Does anyone have any ideas or could show me how the method actually cancels out the non constant coefficeints? thanks. oh i think you are supposed to use liebnitz?? not sure

If you use the anzats and take two derivatives we get

and

Now if you plug this into the homogenious ODE you get

lets reduce to get

Since

Solving we get so

Plug these back into your anzats and you have your complimentry solution